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Non-negative Martingale Solutions to the Stochastic Porous Medium Equation with Sticky Behavior

Ben Hambly, Dörte Kreher, Konstantins Starovoitovs

Abstract

We construct non-negative martingale solutions to the stochastic porous medium equation in one dimension with homogeneous Dirichlet boundary conditions which exhibit a type of sticky behavior at zero. The construction uses the stochastic Faedo--Galerkin method via spatial semidiscretization, so that the pre-limiting system is given by a finite-dimensional diffusion with Wentzell boundary condition. We derive uniform moment estimates for the discrete systems by an Aubin-Lions-type interpolation argument, which enables us to implement a general weak convergence approach for the construction of martingale solutions of an SPDE using a Skorokhod representation-type result for non-metrizable spaces. We rely on a stochastic argument based on the occupation time formula for continuous semimartingales for the identification of the diffusion coefficient in the presence of an indicator function.

Non-negative Martingale Solutions to the Stochastic Porous Medium Equation with Sticky Behavior

Abstract

We construct non-negative martingale solutions to the stochastic porous medium equation in one dimension with homogeneous Dirichlet boundary conditions which exhibit a type of sticky behavior at zero. The construction uses the stochastic Faedo--Galerkin method via spatial semidiscretization, so that the pre-limiting system is given by a finite-dimensional diffusion with Wentzell boundary condition. We derive uniform moment estimates for the discrete systems by an Aubin-Lions-type interpolation argument, which enables us to implement a general weak convergence approach for the construction of martingale solutions of an SPDE using a Skorokhod representation-type result for non-metrizable spaces. We rely on a stochastic argument based on the occupation time formula for continuous semimartingales for the identification of the diffusion coefficient in the presence of an indicator function.

Paper Structure

This paper contains 24 sections, 26 theorems, 312 equations.

Table of Contents

  1. Introduction
  2. Setup and main result
  3. Preliminaries
  4. Notation
  5. Functional setting
  6. Discretization
  7. Setting
  8. Approximate discrete model
  9. Martingale solutions and main result
  10. Moment estimates
  11. Estimates in $L^p(\Omega, L^\infty([0,T], \mathbf H^0_n))$ and $L^{p}(\Omega, L^2([s,t], \mathbf H^1_n))$
  12. Estimates in $L^p(\Omega, L^\infty([0,T], \mathbf H^{(\alpha-2)/\alpha}_n))$
  13. Estimates in $L^p(\Omega, W^{s,p}([0,T], \mathbf H^{-1/\alpha}_n))$
  14. Estimates in $L^2(\Omega, C^{\gamma\beta_1, \beta_2}_0(\mathcal{Q}_T))$ for $\gamma \in (0,1)$
  15. Convergence of the discrete scheme
  16. ...and 9 more sections

Key Result

Lemma 2.1

For any $s_1, s_2 \in [-1, 1]$, $s_1 \leq s_2$,

Theorems & Definitions (61)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 51 more